Solidification Behavior of Dy-Tb-Fe Alloys through Experimental Study and Thermodynamic Calculation

In this work, the solidification microstructure and phase transitions of Dy-Tb-Fe alloy samples were studied by using scanning electron microscopy with energy dispersive spectroscopy (SEM-EDS), X-ray diffraction (XRD) and differential thermal analysis (DTA). No stable ternary compound was detected in the present experiments. The phase transformation temperatures of eight Dy-Tb-Fe alloy samples were measured. Based on the experimental results determined in this work and reported in the literature, the phase equilibria of the Dy-Tb-Fe system was calculated using the CALPHAD method. The calculated vertical sections are consistent with the experimental results determined in this work and reported in the literature. Furthermore, in combination with the experimental solidification microstructure, the solidification behavior of Dy-Tb-Fe alloy samples was analyzed through the thermodynamic calculation with the Gulliver–Scheil non-equilibrium model. The simulated results agree well with the experimental results. This indicates that the reasonable thermodynamic parameters of the Dy-Tb-Fe system were finally obtained.


Introduction
Nd-Fe-B permanent magnets with excellent magnetic properties have been widely used in various industrial fields such as wind turbines, electric vehicles, and aerospace [1][2][3][4][5]. With the increase in the operation temperature, the magnetic properties (e.g., coercivity and remanence) of Nd-Fe-B permanent magnets decrease [6][7][8][9]. In order to ensure the sufficient coercivity of Nd-Fe-B permanent magnets at the operation temperature, the addition of heavy rare earth elements (Dy, Tb, etc.) to Nd-Fe-B permanent magnets to partially substitute Nd is an effective method because the magnetocrystalline anisotropy fields of Dy 2 Fe 14 B and Tb 2 Fe 14 B are much higher than that of Nd 2 Fe 14 B, which would result in the great improvement of the coercivity for the permanent magnets [10][11][12]. To reduce the content of heavy rare earth metals Dy and Tb in Nd-Fe-B permanent magnets, the grain boundary diffusion process (GBDP) was developed recently [13]. During the GBDP, first, Dy and Tb can diffuse along the grain boundary phase into the interior of the magnet; then, the partial Nd on the surface of Nd 2 Fe 14 B grains is replaced by Dy and Tb to form a core-shell microstructure containing Nd 2 Fe 14 B and (Dy, Tb) 2 Fe 14 B. The substitution of Nd by Dy and Tb dilutes the ferromagnetism of the grain boundary phase and further weakens the coupling between neighboring grains, which enhances the coercivity of the magnets after the grain boundary diffusion process [14,15]. In order to gain a deeper understanding of the grain boundary diffusion process, the thermodynamics and kinetics of Nd-Fe-B permanent magnets containing Tb and Dy elements are fundamental to enhance the overall performance of Nd-Fe-B magnets [16]. In particular, the solidification process The thermodynamic database of Dy-Fe and Tb-Fe systems in the high-temperature range (above 800 K) was established by Landin et al. [32], but the contribution of magnetism to the Gibbs energy and experimental heat capacity of all intermetallic compounds were not taken into account. Rong et al. [22] optimized the Dy-Fe and Tb-Fe systems using the CALPHAD method considering the magnetic contribution and experimental heat capacity of the intermetallic compounds. The calculation results including phase relationship and thermodynamic properties are in good agreement with the experimental results. Recently, Ye et al. [24] re-conducted thermodynamic calculations for Dy-Fe and Tb-Fe systems to achieve compatibility with the RE-Fe thermodynamic database. The results of the Dy-Fe and Tb-Fe systems optimized by Ye et al. [24] were used in the present calculation of the Dy-Tb-Fe system.
The Dy-Tb phase diagram was not calculated in the reported literature up to now. Gschneidner et al. [33,34] reported that Dy and Tb are completely miscible and measured lattice parameters of the Dy-Tb alloys in the systematic review of lanthanide binary systems. According to the reported data [33,34], the Dy-Tb phase diagram was drawn by Moffatt [35] considering the continuous solid solution phases formed from β-Dy, β-Tb, α-Dy and α-Tb phases due to their same crystal structures [36]. Therefore, all the phases including liquid phase, bcc (β-Dy, β-Tb) and hcp (α-Dy, α-Tb) in the Dy-Tb system were described by using the ideal solution model. Figure 1 shows the calculated Dy-Tb phase diagram in this work.

Ternary System
The Dy0.73Tb0.27-Fe vertical section of the Dy-Tb-Fe system was measured by Westwood et al. [37] with DTA, X-ray diffraction and metallography. Landin et al. [32] directly extrapolated the Dy-Tb-Fe system based on the experimental results of Westwood et al. and Abell et al. [38], including the Dy0.73Tb0.27-Fe vertical section and liquidus projection.
Although the values of the calculated Dy0.73Tb0.27-Fe vertical section are consistent with the experimental data [37], Ye et al. [24] have recently reassessed the calculations of the Dy-Fe and Tb-Fe systems. Therefore, thermodynamic calculations for the Dy-Tb-Fe system are still necessary.

Experimental Procedure
Eight Dy-Tb-Fe alloy samples with compositions of : = 1: 1 were prepared from bulk Dy, Tb and Fe (99.99% purity, China New Metal Material Technology Company, Ltd., Beijing, China). Each alloy sample (about 4 g) was melted 3-5 times in a vacuum arc melting furnace filled with argon gas to ensure uniform composition. The alloy samples were cooled in a copper crucible with cooling water, and thus the solidification process of the alloy samples was non-equilibrium due to the fast cooling rate. The as-cast alloy samples were prepared by standard metallographic procedure. The morphology and phase composition of the alloy samples were measured by scanning electron microscope with energy dispersive spectroscopy (SEM-EDS, FEI 450G, FEI Company,USA). The compositions of each phase in the alloy samples were measured four times by EDS, and the standard deviations of the measured composition data were determined. After the alloy sample was ground into powder in anhydrous ethanol, the phase structures of the formed phases in the alloy samples were analyzed by X-ray powder diffraction (XRD, PLXcel 3D, Cu Kα radiation). The phase transition temperatures of the alloy samples were measured by differential thermal analysis (DTA, TA Instruments SDT/Q-600) using high-purity Al2O3 crucibles in a flowing argon atmosphere. Considering that rare earth metals are prone to oxidation, a heating/cooling rate of 20 K/min was used in DTA measurement.

Ternary System
The Dy 0.73 Tb 0.27 -Fe vertical section of the Dy-Tb-Fe system was measured by Westwood et al. [37] with DTA, X-ray diffraction and metallography. Landin et al. [32] directly extrapolated the Dy-Tb-Fe system based on the experimental results of Westwood et al. and Abell et al. [38], including the Dy 0.73 Tb 0.27 -Fe vertical section and liquidus projection.
Although the values of the calculated Dy 0.73 Tb 0.27 -Fe vertical section are consistent with the experimental data [37], Ye et al. [24] have recently reassessed the calculations of the Dy-Fe and Tb-Fe systems. Therefore, thermodynamic calculations for the Dy-Tb-Fe system are still necessary.

Experimental Procedure
Eight Dy-Tb-Fe alloy samples with compositions of x Dy : x Tb = 1 : 1 were prepared from bulk Dy, Tb and Fe (99.99% purity, China New Metal Material Technology Company, Ltd., Beijing, China). Each alloy sample (about 4 g) was melted 3-5 times in a vacuum arc melting furnace filled with argon gas to ensure uniform composition. The alloy samples were cooled in a copper crucible with cooling water, and thus the solidification process of the alloy samples was non-equilibrium due to the fast cooling rate.
The as-cast alloy samples were prepared by standard metallographic procedure. The morphology and phase composition of the alloy samples were measured by scanning electron microscope with energy dispersive spectroscopy (SEM-EDS, FEI 450G, FEI Company, Hillsboro, OR, USA). The compositions of each phase in the alloy samples were measured four times by EDS, and the standard deviations of the measured composition data were determined. After the alloy sample was ground into powder in anhydrous ethanol, the phase structures of the formed phases in the alloy samples were analyzed by X-ray powder diffraction (XRD, PLXcel 3D, Cu K α radiation). The phase transition temperatures of the alloy samples were measured by differential thermal analysis (DTA, TA Instruments SDT/Q-600) using high-purity Al 2 O 3 crucibles in a flowing argon atmosphere. Considering that rare earth metals are prone to oxidation, a heating/cooling rate of 20 K/min was used in DTA measurement.

Solution Phases
The solution phase ϕ including liquid, fcc, bcc and hcp is described by using the substitutional solution model. The molar Gibbs energy of the solution phase ϕ can be expressed as follows: where x i is the mole fraction of element i (i = Dy, Tb, Fe) and 0 G ϕ i means the molar Gibbs energy of phase ϕ for element i (i = Dy, Tb, Fe); these values refer to the SGTE database [39]. R is the gas constant and T is the absolute temperature (Kelvin). mag G ϕ m is the magnetic contribution to Gibbs energy of the magnetic phase. In Equation (

Intermetallic Compounds
In the Dy-Tb-Fe system, a continuous solid solution is formed because of the same crystal structure of DyFe 2 and TbFe 2 [40][41][42]. Similarly, DyFe 3 and TbFe 3 , Dy 6 Fe 23 and Tb 6 Fe 23 , Dy 2 Fe 17 and Tb 2 Fe 17 also form a continuous solid solution in the Dy-Tb-Fe system. Therefore, these intermetallic compounds are modeled by (Dy, Tb) 0.3333 Fe 0.66667 , (Dy, Tb) 0. 25 j L REFe 2 Dy,Tb: in which j L REFe 2 Dy:Fe , j L REFe 2 Tb:Fe , j L Dy,Tb:Fe are the interaction parameters to be optimized.

Results and Discussion
The microstructure and phase transitions of eight Dy-Tb-Fe as-cast alloy samples were determined in this work. The phase compositions and phase transition temperatures of the alloy samples measured by EDS, XRD and DTA are shown in Table 1.   Figure 2a, the microstructure of this sample shows the formation of two phases, and the composition of the gray phase was measured by EDS to be 17.88 at.% Dy, 13.91 at.% Tb and 68.21 at.% Fe, while that of the light gray phase was determined to be 46.77 at.% Dy, 47.74 at.% Tb and 5.49 at.% Fe. According to the results determined by EDS in Table 1, the gray phase and the light gray phase were identified to be (Dy, Tb) Fe 2 and hcp (Dy, Tb), respectively, which was same as those of the XRD patterns in Figure 2b. Moreover, there is a large amount of the hcp (Dy, Tb) phase in Figure 2b, and the background of the diffraction pattern is too high and the spectral peak is not smooth, which is the result of the internal stress or preferred orientation generated during the preparation of metal powders. In addition, the microstructure characteristics of the Dy 42.5 Tb 42.5 Fe 15 alloy sample indicate that the light gray hcp (Dy, Tb) phase was formed first from the liquid phase during the solidification process.
phases, while their XRD patterns demonstrate the formation of the (Dy, Tb) Fe2 and hcp (Dy, Tb) phase in Figure 3b,d. According to EDS results, there are two phases present in these two samples, with (Dy, Tb) Fe2 in dark gray and hcp (Dy, Tb) in light gray. It means that the SEM-EDS results of these two samples are consistent with their XRD results. Meanwhile, the formation of the primary phase (Dy, Tb) Fe2 and similar eutectic microstructure including (Dy, Tb) Fe2 and hcp (Dy, Tb) was observed from the BSE micrographs of these two as-cast alloy samples.   of the internal stress or preferred orientation generated during the preparation of metal powders. In addition, the microstructure characteristics of the Dy42.5Tb42.5Fe15 alloy sample indicate that the light gray hcp (Dy, Tb) phase was formed first from the liquid phase during the solidification process. Figure 3 shows the BSE and XRD images of Dy25Tb25Fe50 and Dy20Tb20Fe60 alloy samples. In Figure 3a,c, Dy25Tb25Fe50 and Dy20Tb20Fe60 alloy samples are composed of two phases, while their XRD patterns demonstrate the formation of the (Dy, Tb) Fe2 and hcp (Dy, Tb) phase in Figure 3b,d. According to EDS results, there are two phases present in these two samples, with (Dy, Tb) Fe2 in dark gray and hcp (Dy, Tb) in light gray. It means that the SEM-EDS results of these two samples are consistent with their XRD results. Meanwhile, the formation of the primary phase (Dy, Tb) Fe2 and similar eutectic microstructure including (Dy, Tb) Fe2 and hcp (Dy, Tb) was observed from the BSE micrographs of these two as-cast alloy samples.   Table 1, these two phases are identified to be (Dy, Tb) Fe 2 and (Dy, Tb) Fe 3 , which are in good agreement with the XRD patterns in Figure 4b  As shown in Figure 4a,c, Dy15Tb15Fe70 and Dy13.5Tb13.5Fe73 alloy samples present the microstructure of two phases. Based on the composition measurements in Table 1, these two phases are identified to be (Dy, Tb) Fe2 and (Dy, Tb) Fe3, which are in good agreement with the XRD patterns in Figure 4b,d. Similarly, the microstructure characteristics of Dy15Tb15Fe70 and Dy13.5Tb13.5Fe73 alloy samples show that the (Dy, Tb) Fe3 phase is the primary phase during this solidification process.  Figure 5 displays the BSE micrographs and XRD patterns of Dy12Tb12Fe76 and Dy10.5Tb10.5Fe79 alloy samples. In Figure 5a,c, three different phases were formed in these two samples. As given in Table 1, the experimental results obtained by EDS indicate that the light gray phase, the gray phase, and the dark black phase are (Dy, Tb) Fe3, (Dy, Tb)6 Fe23 and (Dy, Tb)2 Fe17, respectively, which are same as those of the XRD patterns in Figure  5b,d. The microstructures of Dy12Tb12Fe76 and Dy10.5Tb10.5Fe79 alloy samples suggest that the (Dy, Tb)6Fe23 phase as the primary phase was formed. Figure 6 presents the BSE and XRD images of the Dy9Tb9Fe82 alloy sample. Based on the microstructure and XRD patterns in Figure 6a,b with the phase compositions measured by EDS in Table 1, the formation of the (Dy, Tb)6 Fe23 phase and the (Dy, Tb)2 Fe17 phase was found. Moreover, (Dy, Tb)2 Fe17 was formed as the primary phase in the Dy9Tb9Fe82 alloy sample.   Figure 5a,c, three different phases were formed in these two samples. As given in Table 1, the experimental results obtained by EDS indicate that the light gray phase, the gray phase, and the dark black phase are (Dy, Tb) Fe 3 , (Dy, Tb) 6 Fe 23 and (Dy, Tb) 2 Fe 17 , respectively, which are same as those of the XRD patterns in Figure 5b  Based on the experimental results of eight Dy-Tb-Fe alloy samples with the compositions ofx : x = 1: 1 determined by SEM-EDS and XRD, it was noted that the stable ternary intermetallic compound was not determined in this work. In addition, the EDS results measured in Table 1 illustrate that the solubilities of Dy in TbFe2, TbFe3, Tb6Fe23 and Tb2Fe17 as well as those of Tb in DyFe2, DyFe3, Dy6Fe23 and Dy2Fe17 are different. This indicates that (Dy, Tb) Fe2, (Dy, Tb) Fe3, (Dy, Tb)6Fe23 and (Dy, Tb)2 Fe17 all form continuous solid solution phase in the Dy-Tb-Fe system, which was also reported by Westwood et al. [37]. Figure 7 shows the thermal analysis curve of Dy-Tb-Fe alloy samples in this work. Based on the thermal analysis results, the transition temperatures of Dy-Tb-Fe alloy samples were analyzed, and the results are listed in Table 1. In Figure 7a, the thermal curve of the Dy42.5Tb42.5Fe15 alloy sample displays three peaks at 1030 K, 1057 K and 1435 K.  Based on the experimental results of eight Dy-Tb-Fe alloy samples with the compositions ofx : x = 1: 1 determined by SEM-EDS and XRD, it was noted that the stable ternary intermetallic compound was not determined in this work. In addition, the EDS results measured in Table 1 illustrate that the solubilities of Dy in TbFe2, TbFe3, Tb6Fe23 and Tb2Fe17 as well as those of Tb in DyFe2, DyFe3, Dy6Fe23 and Dy2Fe17 are different. This indicates that (Dy, Tb) Fe2, (Dy, Tb) Fe3, (Dy, Tb)6Fe23 and (Dy, Tb)2 Fe17 all form continuous solid solution phase in the Dy-Tb-Fe system, which was also reported by Westwood et al. [37]. Figure 7 shows the thermal analysis curve of Dy-Tb-Fe alloy samples in this work. Based on the thermal analysis results, the transition temperatures of Dy-Tb-Fe alloy samples were analyzed, and the results are listed in Table 1. In Figure 7a, the thermal curve of the Dy42.5Tb42.5Fe15 alloy sample displays three peaks at 1030 K, 1057 K and 1435 K. Based on the experimental results of eight Dy-Tb-Fe alloy samples with the compositions of x Dy : x Tb = 1 : 1 determined by SEM-EDS and XRD, it was noted that the stable ternary intermetallic compound was not determined in this work. In addition, the EDS results measured in Table 1 illustrate that the solubilities of Dy in TbFe 2 , TbFe 3 , Tb 6 Fe 23 and Tb 2 Fe 17 as well as those of Tb in DyFe 2 , DyFe 3 , Dy 6 Fe 23 and Dy 2 Fe 17 are different. This indicates that (Dy, Tb) Fe 2 , (Dy, Tb) Fe 3 , (Dy, Tb) 6 Fe 23 and (Dy, Tb) 2 Fe 17 all form continuous solid solution phase in the Dy-Tb-Fe system, which was also reported by Westwood et al. [37]. were analyzed, and the results are listed in Table 1. In Figure 7a, the thermal curve of the Dy 42.5 Tb 42.5 Fe 15 alloy sample displays three peaks at 1030 K, 1057 K and 1435 K. Combined with Figure 2a, the peaks at 1030 K and 1057 K are generated due to the formation of the hcp (Dy, Tb) phase and the (Dy, Tb) Fe 2 phase, while the third peak at 1435 K is corresponding to the formation of the hcp (Dy, Tb) phase as the primary phase. The thermal curve of the Dy 25 Tb 25 Fe 50 alloy sample in Figure 7b presents three signal peaks at 1121 K, 1131 K and 1459 K, respectively, which correspond to the formation of the (Dy, Tb) Fe 2 and the hcp (Dy, Tb) phase. Nevertheless, the thermal curve of the Dy 20 Tb 20 Fe 60 alloy sample in Figure 7c shows only one endothermic peak at 1490 K, corresponding to the formation of (Dy, Tb) Combined with Figure 2a, the peaks at 1030 K and 1057 K are generated due to the formation of the hcp (Dy, Tb) phase and the (Dy, Tb) Fe2 phase, while the third peak at 1435 K is corresponding to the formation of the hcp (Dy, Tb) phase as the primary phase. The thermal curve of the Dy25Tb25Fe50 alloy sample in Figure 7b presents three signal peaks at 1121 K, 1131 K and 1459 K, respectively, which correspond to the formation of the (Dy, Tb) Fe2 and the hcp (Dy, Tb) phase. Nevertheless, the thermal curve of the Dy20Tb20Fe60 alloy sample in Figure 7c shows only one endothermic peak at 1490 K, corresponding to the formation of (Dy, Tb) Fe2, although the solidification microstructures of both Dy25Tb25Fe50 and Dy20Tb20Fe60 alloy samples contain the (Dy, Tb) Fe2 and hcp (Dy, Tb) phases. In Figure 7d,e, the thermal analysis curves of Dy15Tb15Fe70 and Dy13.5Tb13.5Fe73 alloy samples indicate two peaks at 1494/1475 K and 1501/1505 K, corresponding to the formation of (Dy, Tb) Fe2 and (Dy, Tb) Fe3. Similarly, the thermal analysis curve of the Dy12Tb12Fe76 alloy sample in Figure 7f indicates three signal peaks at 1482 K, 1495 K and 1555 K, corresponding to the formation of (Dy, Tb) Fe3, (Dy, Tb)6 Fe23 and (Dy, Tb)2 Fe17. However, two peaks at 1497 K and 1562 K were observed in the DTA results of the Dy10.5Tb10.5Fe79 alloy sample in Figure 7g, although the solidification microstructures of both Dy12Tb12Fe76 and Dy10.5Tb10.5Fe79 alloy samples consist of (Dy, Tb) Fe3, (Dy, Tb)6 Fe23 and (Dy, Tb)2 Fe17. As shown in Figure 7h, two endothermic peaks at 1523 K and 1560 K were observed in the Dy9Tb9Fe82 alloy sample, which correspond to the formation of (Dy, Tb)6 Fe23 and (Dy, Tb)2 Fe17. In addition, the oxidation peaks due to the easy oxidation of Dy-Tb-Fe alloy samples at high temperatures were also observed in Figure 7f,h.

Thermodynamic Calculation
Based on the experimental data determined in this work and reported by Westwood et al. [37] as well as previous evaluations of Dy-Fe, Tb-Fe, and Tb-Dy systems, the Dy-Tb-Fe system was calculated. The thermodynamic parameters of the Dy-Tb-Fe system obtained are listed in Table 2.

Thermodynamic Calculation
Based on the experimental data determined in this work and reported by Westwood et al. [37] as well as previous evaluations of Dy-Fe, Tb-Fe, and Tb-Dy systems, the Dy-Tb-Fe system was calculated. The thermodynamic parameters of the Dy-Tb-Fe system obtained are listed in Table 2.     Figure 8 shows the calculated liquidus projection of the Dy-Tb-Fe system. It was found that no invariant reactions are existent in this ternary system. The calculated liquidus projection agrees with the experimental results of the primary phase determined in the experiment. Figure 9 shows the vertical sections of Tb 0.50 Dy 0.50 -Fe and Tb 0.27 Dy 0.73 -Fe calculated based on the experimental data determined in this work and reported by Westwood et al. [37]. The calculation results differ slightly from the experimental results and are still accepted within the experimental errors considering the oxidation of Dy-Tb-Fe alloy samples in the thermal analysis measurements at high temperature. = −4000 Figure 8 shows the calculated liquidus projection of the Dy-Tb-Fe sy found that no invariant reactions are existent in this ternary system. The cal dus projection agrees with the experimental results of the primary phase d the experiment. Figure 9 shows the vertical sections of Tb0.50Dy0.50-Fe and T calculated based on the experimental data determined in this work and repo wood et al. [37]. The calculation results differ slightly from the experimenta are still accepted within the experimental errors considering the oxidation alloy samples in the thermal analysis measurements at high temperature.  In order to gain a deeper understanding of the phase transformation of Dy-Tb-F alloy samples during the non-equilibrium solidification process, the thermodynamic pa rameters of the Dy-Tb-Fe system obtained in this work were employed to simulate th solidification process of the as-cast alloy samples using the Gulliver-Scheil model. Figur 10 displays the solidification process of four alloy samples (Dy42.5Tb42.5Fe15, Dy15Tb15Fe7 Dy10.5Tb10.5Fe79, and Dy9Tb9Fe82) simulated using the Gulliver-Scheil module. The simu lated solidification path of the Dy42.5Tb42.5Fe15 alloy sample in Figure 10a is as follows: L → hcp (Dy, Tb) + L → hcp (Dy, Tb) +L + (Dy, Tb) Fe2. The calculated results show that th solidification structure of the Dy42.5Tb42.5Fe15 alloy sample consists of a hcp (Dy, Tb) phas and (Dy, Tb) Fe2, which is consistent with the microstructure observation results show in Figure 2a. Figure 10b shows the simulated solidification path of the Dy15Tb15Fe70 allo sample: L → (Dy, Tb) Fe3 + L → (Dy, Tb) Fe3 + L + (Dy, Tb) Fe2. It indicates that the micro structure of the Dy15Tb15Fe70 alloy sample consists of (Dy, Tb) Fe3 and (Dy, Tb) Fe2. Th simulation results are the same as the SEM results shown in Figure 4a. Similarly, Figur  10c shows the simulated solidification path of the Dy10.5Tb10.5Fe79 alloy sample: L → L (Dy, Tb)6Fe23 → L + (Dy, Tb)6 Fe23 + (Dy, Tb) Fe3, while the simulated solidification path o the Dy9Tb9Fe82 alloy sample in Figure 10d is as follows: L → L + (Dy, Tb)2Fe17 → L + (Dy Tb)2Fe17 + (Dy, Tb)6Fe23, which is consistent with the microstructure observation resul shown in Figures 5c and 6a. This indicates that reliable thermodynamic data of the Dy Tb-Fe system was obtained in this work, which can be employed to reproduce well th solidification processes of Dy-Tb-Fe alloy samples using the Gulliver-Scheil module. In order to gain a deeper understanding of the phase transformation of Dy-Tb-Fe alloy samples during the non-equilibrium solidification process, the thermodynamic parameters of the Dy-Tb-Fe system obtained in this work were employed to simulate the solidification process of the as-cast alloy samples using the Gulliver-Scheil model. Figure Figure 4a. Similarly, Figure 10c shows the simulated solidification path of the Dy 10.5 Tb 10.5 Fe 79 alloy sample: L → L + (Dy, Tb) 6 Fe 23 → L + (Dy, Tb) 6 Fe 23 + (Dy, Tb) Fe 3 , while the simulated solidification path of the Dy 9 Tb 9 Fe 82 alloy sample in Figure 10d is as follows: L → L + (Dy, Tb) 2 Fe 17 → L + (Dy, Tb) 2 Fe 17 + (Dy, Tb) 6 Fe 23 , which is consistent with the microstructure observation results shown in Figures 5c and 6a. This indicates that reliable thermodynamic data of the Dy-Tb-Fe system was obtained in this work, which can be employed to reproduce well the solidification processes of Dy-Tb-Fe alloy samples using the Gulliver-Scheil module.

Conclusions
This work investigates the solidification behavior of the Dy-Tb-Fe system using experimental measurements and thermodynamic calculations. The conclusions drawn are as follows: 1. The phase transition temperatures and phase compositions of eight Dy-Tb-Fe alloy samples were determined. Based on the experimental results determined in this work and reported in the literature, the thermodynamic calculation of the Dy-Tb-Fe system was performed using the CALPHAD method. The calculated vertical section and liquidus projection are consistent with the experimental results. 2. The solidification behaviors of several Dy-Tb-Fe alloy samples were simulated by using the Gulliver-Scheil non-equilibrium model with the obtained thermodynamic parameters. The simulation results were compared with the solidification structure of the experimental samples, and they were in good agreement with the experimental results. This means that the thermodynamic parameters of the Dy-Tb-Fe system optimized in this work are reliable and will provide reference for the microstructure design of high-performance Nd-Dy-Tb-Fe-B magnets. Author

Conclusions
This work investigates the solidification behavior of the Dy-Tb-Fe system using experimental measurements and thermodynamic calculations. The conclusions drawn are as follows: 1.
The phase transition temperatures and phase compositions of eight Dy-Tb-Fe alloy samples were determined. Based on the experimental results determined in this work and reported in the literature, the thermodynamic calculation of the Dy-Tb-Fe system was performed using the CALPHAD method. The calculated vertical section and liquidus projection are consistent with the experimental results.

2.
The solidification behaviors of several Dy-Tb-Fe alloy samples were simulated by using the Gulliver-Scheil non-equilibrium model with the obtained thermodynamic parameters. The simulation results were compared with the solidification structure of the experimental samples, and they were in good agreement with the experimental results. This means that the thermodynamic parameters of the Dy-Tb-Fe system optimized in this work are reliable and will provide reference for the microstructure design of high-performance Nd-Dy-Tb-Fe-B magnets. Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.

Data Availability Statement:
All the data that support the findings of this study are included within the article.